Random numbers from the tails of probability distributions using the transformation method

TL;DR

This paper introduces a novel transformation method for efficiently generating random numbers from the tails of various probability distributions, enhancing existing techniques like rejection sampling for distributions with infinite support.

Contribution

The authors develop a new method to quickly generate tail and interval-specific random numbers, improving upon traditional transformation and rejection sampling approaches.

Findings

Method effectively generates tail random numbers for Levy and Mittag-Leffler distributions.

Demonstrates improved sampling efficiency in stochastic solutions of fractional diffusion equations.

Validates the method's accuracy through properties of transform maps.

Abstract

The speed of many one-line transformation methods for the production of, for example, Levy alpha-stable random numbers, which generalize Gaussian ones, and Mittag-Leffler random numbers, which generalize exponential ones, is very high and satisfactory for most purposes. However, for the class of decreasing probability densities fast rejection implementations like the Ziggurat by Marsaglia and Tsang promise a significant speed-up if it is possible to complement them with a method that samples the tails of the infinite support. This requires the fast generation of random numbers greater or smaller than a certain value. We present a method to achieve this, and also to generate random numbers within any arbitrary interval. We demonstrate the method showing the properties of the transform maps of the above mentioned distributions as examples of stable and geometric stable random numbers used for the stochastic solution of the space-time fractional diffusion equation.